Natural modes of Bernoulli–Euler beams with symmetric cracks

The equation of motion and associated boundary conditions are derived for a uniform Bernoulli-Euler beam containing one single-edge crack. The main idea is to use a generalized vari-ational principle that allows for modified stress, strain, and displacement fields that enable one to satisfy the compatibility requirements in the vicinity of the crack. The concentration in stress is represented by introducing a crack function into the beam's compatibility relations. A displacement function is also introduced to modify the in-plane displacement and its slope near the crack. Both functions are chosen to have their maximum value at the ' cracked section and to decay ex-ponentially along the beam's longitudinal direction. The rate of exponential decay is evaluated from finite element calcula-tions. The resulting equation of motion is solved for simply supported and cantilevered beams with single-edge cracks by a Gderkin and a local Ritz procedure, respectively. These theo-retical natural frequencies and mode shapes are confirmed by comparisons with experimental and finite element results, and in both cases a close match is obtained. The possibility of determining the cracked beams ' damage properties from the changes of its dynamic behavior is discussed. Nomenclature crack depth ith generalized coordinate amplitude beam cross section area half breadth of rectangular beam: &, crack ratio half depth of rectangular beam Young's modulus of elasticity crack function a $, frequency ratio unit step function cross-sectional area moment of interia element stiffness matrix global stiffness matrix a SA z f d A a SA zcpdA G SA zcptd